An important property satisfied by \(\mathrm {THH}\)  of ring spectra is Morita invariance: if two ring spectra \(R\)  and \(R^{\prime }\)  are Morita equivalent (i.e., their model categories of modules are Quillen equivalent), then \(\mathrm {THH}(R) \simeq \mathrm {THH}(R^{\prime })\)  [1]}. In line with the idea that a shadow is an axiomatic, bicategorical generalization of \(\mathrm {THH}\) , Campbell and Ponto proved that shadows satisfied a natural generalization of Morita invariance [2]}. In this section we recover Morita invariance of shadows using the machinery we developed in the previous sections, which enables us to define a functorial lift of the useful Euler characteristic construction of Campbell and Ponto, whose work we begin by reviewing.
